A survey of Wall’s finiteness obstruction. (English) Zbl 0967.57003

Ferry, Steve; Ranicki, Andrew

Cappell, Sylvain (ed.) et al., Surveys on surgery theory. Vol. 2: Papers dedicated to C. T. C. Wall on the occasion of his 60th birthday. Princeton, NJ: Princeton University Press. Ann. Math. Stud. 149, 63-79 (2001).

Introduction: Wall’s finiteness obstruction is an algebraic \(K\)-theory invariant which decides if a finitely dominated space is homotopy equivalent to a finite CW complex. The invariant was originally formulated in the context of surgery on CW complexes, generalizing Swan’s application of algebraic \(K\)-theory to the study of free actions of finite groups on spheres. In the context of surgery on manifolds, the invariant first arose as the Siebenmann obstruction to closing a tame end of a non-compact manifold. The object of this survey is to describe the Wall finiteness obstruction and some of its many applications to the surgery classification of manifolds. The book of K. Varadarajan [The finiteness obstruction of C. T. C. Wall (1989; Zbl 0753.57002)] and the survey of G. Mislin [in “Handbook of algebraic topology”, 1259-1291 (1995; Zbl 0870.57030)] deal with the finiteness obstruction from a more homotopy theoretic point of view.
For the entire collection see [Zbl 0957.00062].

MSC:

57-02	Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
19J25	Surgery obstructions (\(K\)-theoretic aspects)
57R65	Surgery and handlebodies
57R67	Surgery obstructions, Wall groups

Keywords:

algebraic \(K\)-theory

Citations:

Zbl 0753.57002; Zbl 0870.57030